arXiv subjects

Maleeha Khawaja

Publications and source records attributed to Maleeha Khawaja.

11 recordsLinked to original sources

arXiv2026

Conductor exponents for families of hyperelliptic curves

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families.

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arXiv2026

On the unit equation $\varepsilon+δ=n$ in cubic fields

Let $n$ be an integer not equal to $-2$, $0$ or $2$. We consider the unit equation $\varepsilon + δ= n$ in units $\varepsilon, δ$ of cubic fields. We show that this equation has no solutions for 100% of cubic fields, when ordered by discriminant. This is consistent with a recent conjecture of the authors.

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arXiv2026

Primitive points on some low degree Fermat curves

Let $n\geq 3$ be an integer. Let $F_n$ be the Fermat curve defined by the Fermat equation $x^n+y^n=z^n$. For a curve $C/\mathbb{Q}$, we say an algebraic point $P\in C(\bar{\mathbb{Q}})$ is primitive if the Galois group of the Galois closure of the number field $\mathbb{Q}(P)$ is a primitive permutation group. Recall that $A_4$ is a primitive subgroup of $S_4$. We prove that there are no non-trivial quartic points on $F_n$ with Galois closure $A_4$, when $n = 7$ and $n = 8$. We also provide sufficient conditions for the non-existence of non-trivial points on the Fermat curves $F_6$ and $F_8$ defined over a given primitive number field of degree at least $3$.

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arXiv2025

New Algebraic Points on Curves

Let $C$ be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field $L$, we define the set of $L$-new points on $C$ to be $C(L)_{new} = \{P \in C(L) : \mathbb{Q}(P)=L\}$; this is the set of points on $C$ defined over $L$ but not any strictly smaller field. Let $n$ be at least 2. We conjecture that $C(L)_{new}$ is empty for 100 percent of degree $n$ number fields $L$ when ordered by absolute discriminant. For degrees $n=2$, $3$, we give sufficient criteria for our conjecture to hold in terms of an explicit model for $C$. For general $n$ we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for $n=2$ and $C=X_0(N)$ for the $18$ values $N \ne 37$ such that $X_0(N)$ is hyperelliptic, and also for $n=3$ and $C=X_0(23)$, $X_0(29)$, $X_0(31)$, $X_0(64)$. Moreover, we prove the analogue of our conjecture for the unit equation, again with $n=3$.

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arXiv2024

Primitive algebraic points on curves

A number field $K$ is primitive if $K$ and $\mathbb{Q}$ are the only subextensions of $K$. Let $C$ be a curve defined over $\mathbb{Q}$. We call an algebraic point $P\in C(\overline{\mathbb{Q}})$ primitive if the number field $\mathbb{Q}(P)$ is primitive. We present several sets of sufficient conditions for a curve $C$ to have finitely many primitive points of a given degree $d$. For example, let $C/\mathbb{Q}$ be a hyperelliptic curve of genus $g$, and let $3 \le d \le g-1$. Suppose that the Jacobian $J$ of $C$ is simple. We show that $C$ has only finitely many primitive degree $d$ points, and in particular it has only finitely many degree $d$ points with Galois group $S_d$ or $A_d$. However, for any even $d \ge 4$, a hyperelliptic curve $C/\mathbb{Q}$ has infinitely many imprimitive degree $d$ points whose Galois group is a subgroup of $S_2 \wr S_{d/2}$.

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arXiv2024

Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$

In this paper, we begin the study of the Fermat equation $x^n+y^n=z^n$ over real biquadratic fields. In particular, we prove that there are no non-trivial solutions to the Fermat equation over $\mathbb{Q}(\sqrt{2},\sqrt{3})$ for $n\geq 4$.

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arXiv2024

Torsion primes for elliptic curves over degree 8 number fields

Let $d\geq 1$ be an integer and let $p$ be a rational prime. Recall that $p$ is a torsion prime of degree $d$ if there exists an elliptic curve $E$ over a degree $d$ number field $K$ such that $E$ has a $K$-rational point of order $p$. Derickx, Kamienny, Stein and Stoll have computed the torsion primes of degrees 4, 5, 6 and 7; we verify that these techniques can be extended to determine the torsion primes of degree 8.

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arXiv2024

The modular approach to Diophantine equations over totally real fields

Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine equations over totally real number fields. We give an extensive overview of these obstacles as well as providing a survey of existing methods and results in this area.

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arXiv2024

A single source theorem for primitive points on curves

Let $C$ be a curve defined over a number field $K$ and write $g$ for the genus of $C$ and $J$ for the Jacobian of $C$. Let $n \ge 2$. We say that an algebraic point $P \in C(\overline{K})$ has degree $n$ if the extension $K(P)/K$ has degree $n$. By the Galois group of $P$ we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$. We say that $P$ is primitive if its Galois group is primitive as a subgroup of $S_n$. We prove the following 'single source' theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$. Suppose that either $J$ is simple, or that $J(K)$ is finite. Suppose $C$ has infinitely many primitive degree $n$ points. Then there is a degree $n$ morphism $φ: C \rightarrow \mathbb{P}^1$ such that all but finitely many primitive degree $n$ points correspond to fibres $φ^{-1}(α)$ with $α\in \mathbb{P}^1(K)$. We prove moreover, under the same hypotheses, that if $C$ has infinitely many degree $n$ points with Galois group $S_n$ or $A_n$, then $C$ has only finitely many degree $n$ points of any other primitive Galois group. The proof makes essential use of recent results of Burness and Guralnick on fixed point ratios of faithful, primitive group actions.

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arXiv2023

On perfect powers that are sums of cubes of a nine term arithmetic progression

We study the equation $(x-4r)^3 + (x-3r)^3 + (x-2r)^3+(x-r)^3 + x^3 + (x+r)^3+(x+2r)^3 + (x+3r)^3 + (x+4r)^3 = y^p$, which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions $0 < r \leq 10^6$, $p \geq 5 $ a prime and $\gcd(x, r) = 1$, we show that solutions must satisfy $xy=0$. Moreover, we study the equation for prime exponents $2$ and $3$ in greater detail. Under the assumptions $r>0$ a positive integer and $\gcd(x, r) = 1$ we show that there are infinitely many solutions for $p=2$ and $p=3$ via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier's Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.

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arXiv2023

Power values of power sums: a survey

Research on power values of power sums has gained much attention of late, partially due to the explosion of refinements in multiple advanced tools in (computational) Number Theory in recent years. In this survey, we present the key tools and techniques employed thus far in the (explicit) resolution of Diophantine problems, as well as an overview of existing results. We also state some open problems that naturally arise in the process.

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